Problem: Yochanan walked from home to the bus stop at an average speed of $5$ $\text{km/h}$. He immediately got on his school bus and traveled at an average speed of $60$ $\text{km/h}$ until he got to school. The total distance from his home to school is $35$ kilometers, and the entire trip took $1.5$ hours. How many kilometers did Yochanan cover by walking, and how many kilometers did he cover by travelling on the bus? Yochanan walked
Answer: Let $x$ represent the time (in hours) Yochanan walked and let $y$ represent the time (in hours) he traveled by bus. Since we have two unknowns, we need two equations to find them. Let's use the given information in order to write two equations containing $x$ and $y$. For instance, we are given that Yochanan walked at an average speed of $\textit{5 km/h}$, the school bus traveled at an average speed of $\textit{60 km/h}$, and the distance between Yochanan's home and his school is $\textit{35}$ kilometers. How can we model this sentence algebraically? The total distance Yochanan walked can be modeled by $5x$, and the total distance he traveled by bus can be modeled by $60y$. Since these together add up to $35$, we get the following equation: $5 x+60 y = 35$ We are also given that the entire trip took $\textit{1.5}$ hours. This can be expressed as: $x + y =1.5$ Now that we have a system of two equations, we can go ahead and solve it! We can now solve the system of equations by the elimination method. Note that the coefficient of $x$ in the first equation, $5$, is exactly $5$ times the coefficient of $x$ in the second equation, $1$. Therefore, we can multiply the second equation by ${-5}$ in order to eliminate $x$. $\begin{aligned} {-5}\cdot x+({-5})\cdot y&={-5}\cdot1.5\\\\ -5x-5y&=-7.5\end{aligned}$ Now we can eliminate $x$ : $\begin{aligned}5x+{60y}&=35\\\\ {+}\ -5x-{5y}&=-7.5\\ \hline\\ 0+55y &=27.5 \end{aligned}$ When we solve the resulting equation, we find that $y = 0.5$, which we can substitute into $x+y=1.5$ to find that $x=1$. Recall that $5x$ denotes the distance Yochanan walked and $60y$ denotes the distance he traveled by bus. Therefore, Yochanan walked $\textit{5}\cdot\textit{1}=\textit{5}$ kilometers and traveled $\textit{60}\cdot\textit{0.5}=\textit{30}$ kilometers by bus.